X-ray Absorption Linear Dichroism at the Ti K-edge of TiO2 anatase single crystals

TL;DR

This study investigates the X-ray absorption linear dichroism at the Ti K-edge in anatase TiO2 single crystals, revealing detailed electronic transition mechanisms and providing insights for characterizing related materials.

Contribution

The paper provides a detailed assignment of pre-edge peaks in Ti K-edge XAS of anatase TiO2 using ab initio calculations and tensor analysis, clarifying the nature of electronic transitions.

Findings

A1 peak is mainly an on-site 3d-4p hybridized transition.

A3 and B peaks are non-local, with A3 being dipolar and B involving longer-range interactions.

A2 exhibits quadrupolar angular dependence, indicating complex transition behavior.

Abstract

Anatase TiO2 (a-TiO2) exhibits a strong X-ray absorption linear dichroism with the X-ray incidence angle in the pre-edge, the XANES and the EXAFS at the titanium K-edge. In the pre-edge region the behaviour of the A1-A3 and B peaks, originating from the 1s-3d transitions, is due to the strong $p$-orbital polarization and strong $p-d$ orbital mixing. An unambiguous assignment of the pre-edge peak transitions is made in the monoelectronic approximation with the support of ab initio finite difference method calculations and spherical tensor analysis in quantitative agreement with the experiment. It is found that A1 is mostly an on-site 3d-4p hybridized transition, while peaks A3 and B are non-local transitions, with A3 being mostly dipolar and influence by the 3d-4p intersite hybridization, while B is due to interactions at longer range. Finally, peak A2 which was previously assigned to a transition involving pentacoordinated titanium atoms exhibits a quadrupolar angular evolution with incidence angle. These results pave the way to the use of the pre-edge peaks at the K-edge of a-TiO2 to characterize the electronic structure of related materials and in the field of ultrafast XAS where the linear dichroism can be used to compare the photophysics along different axes.

Tables

Table 1. Table 1: Previous assignments of the final states involved in the pre-edge transitions of the \ch Ti K-edge XAS spectrum of a- \ch TiO2. The orbitals with dominant contribution to the transition are emphasized in bold. E ​ 1 𝐸 1 E1 is for dipolar transitions and E ​ 2 𝐸 2 E2 for quadrupolar transitions. Off-site transitions are in red, on-site transitions are in black. Irreducible representations corresponding to the final state in the considered point group of a- \ch TiO2 are in parentheses. The dash ”-” symbol refers to orbital hybridization.

reference	A1	A2	A3	B
[10]	$\mathrm{𝟑}{𝐝}_{{𝐱}^{\mathrm{𝟐}}-{𝐲}^{\mathrm{𝟐}}}\left(b_1\right)$, $4p_x,4p_y,3d_{xz},3d_{yz}\left(e\right)$	$\mathrm{𝟒}{𝐩}_{𝐳},\mathrm{𝟑}{𝐝}_{\mathrm{𝐱𝐲}}\left(b_2\right)$, $4p_x,4p_y,3d_{xz},3d_{yz}\left(e\right)$	$\mathrm{𝟒}{𝐩}_{𝐳},\mathrm{𝟑}{𝐝}_{\mathrm{𝐱𝐲}}\left(b_2\right)$, $3d_{z^2}\left(a_1\right)$	$\mathrm{𝟒}𝐩,\mathrm{𝟒}𝐬$
[17]	$E1$: $𝐩\left({𝐭}_{\mathrm{𝟐}𝐠}\right)$, $E2$: $3d\left(t_{2g}\right)$		$E1$: $𝐩\left({𝐞}_{𝐠}\right)$, $𝐩-\mathrm{𝟑}𝐝\left({𝐭}_{\mathrm{𝟐}𝐠}\right)$, $E2$: $3d\left(e_g\right)$	$E1$: ${𝐩}_{𝐳},\mathrm{𝟑}𝐝\left({𝐞}_{𝐠}\right)$
[15]	$E1$: $\mathrm{𝟒}𝐩-\mathrm{𝟑}𝐝\left({𝐭}_{\mathrm{𝟐}𝐠}\right)$, $E2$: $3d\left(t_{2g}\right)$		$E1$: $\mathrm{𝟒}𝐩$,${𝐝}_{\mathrm{𝐱𝐲}}$,${𝐝}_{\mathrm{𝐱𝐳}}$,${𝐝}_{\mathrm{𝐲𝐳}}\left({𝐭}_{\mathrm{𝟐}𝐠}\right)$	$E1$: $\mathrm{𝟒}𝐩$,${𝐝}_{{𝐱}^{\mathrm{𝟐}}-{𝐲}^{\mathrm{𝟐}}}$,${𝐝}_{{𝐳}^{\mathrm{𝟐}}}\left({𝐞}_{𝐠}\right)$
This work	$E1$: $\mathrm{𝟒}{𝐩}_{𝐱,𝐲}-\mathrm{𝟑}{𝐝}_{\mathrm{𝐱𝐳}}$,$\mathrm{𝟑}{𝐝}_{\mathrm{𝐲𝐳}}$, $E2$: $d_{xz}$,$d_{yz}$,$d_{x^2-y^2}$	$E2$: $\mathrm{𝟑}{𝐝}_{\mathrm{𝐱𝐲}}$, $3d_{z^2}$	$E1$: $\mathrm{𝟒}{𝐩}_{𝐱}$,$\mathrm{𝟒}{𝐩}_{𝐲}$,$\mathrm{𝟒}{𝐩}_{𝐳}$–$3d_{xy},3d_{z^2}$	$E1$: $\mathrm{𝟒}{𝐩}_{𝐱}$,$\mathrm{𝟒}{𝐩}_{𝐲}$,$\mathrm{𝟒}{𝐩}_{𝐳}$

Table 2. Table 2: Angular dependence along θ 𝜃 \theta and ϕ italic-ϕ \phi of the dipole and quadrupole XAS cross-section dominant terms in the expansions ( 1 ) and ( 2 ) at the \ch Ti K-edge of a- \ch TiO2 according to the final state of the transition. Transitions to p 𝑝 p final states are dipole allowed ( σ D ​ ( ϵ ^ ) superscript 𝜎 𝐷 ^ italic-ϵ \sigma^{D}(\hat{\epsilon}) ) while transitions to d 𝑑 d final states are quadrupole allowed ( σ Q ​ ( ϵ ^ , k ^ ) superscript 𝜎 𝑄 ^ italic-ϵ ^ 𝑘 \sigma^{Q}(\hat{\epsilon},\hat{k}) ).

final state	${\sigma }^D\left(\widehat{ϵ}\right)$ or ${\sigma }^Q(\widehat{ϵ},\hat{k})$ $\theta$-dependence	${\sigma }^D\left(\widehat{ϵ}\right)$ or ${\sigma }^Q(\widehat{ϵ},\hat{k})$ $\varphi$-dependence
$p_x,p_y$	$-{\cos }^2\theta$	no dependence
$p_z$	${\cos }^2\theta$	no dependence
$d_{z^2}$	${\sin }^2\theta \cdot {\cos }^2\theta$	no dependence
$d_{xy}$	${\sin }^2\theta \cdot {\cos }^2\theta$	$\cos \left(4\varphi \right)$
$d_{x^2-y^2}$	${\sin }^2\theta \cdot {\cos }^2\theta$	$-\cos \left(4\varphi \right)$
$d_{xz}$,$d_{yz}$	$-{\sin }^2\theta \cdot {\cos }^2\theta$	no dependence

Table 3. Table 3: Table of dipolar spherical tensor elements based on the final state reached by the transition. Each element of the table must be multiplied by α ​ ℏ ​ ω ​ δ ​ ( E f − E i − ℏ ​ ω ) ​ ξ f 2 𝛼 Planck-constant-over-2-pi 𝜔 𝛿 subscript 𝐸 𝑓 subscript 𝐸 𝑖 Planck-constant-over-2-pi 𝜔 superscript subscript 𝜉 𝑓 2 \alpha\hbar\omega\delta(E_{f}-E_{i}-\hbar\omega)\xi_{f}^{2} where f 𝑓 f is the final state and ξ f subscript 𝜉 𝑓 \xi_{f} represents the integration of the radial parts of the wavefunctions.

final state	${\sigma }^D(0,0)$	${\sigma }^D(2,0)$
$p_x$	$2/\sqrt{5}$	$2/\sqrt{14}$
$p_y$	$2/\sqrt{5}$	$2/\sqrt{14}$
$p_z$	$2/\sqrt{5}$	$-2/\sqrt{14}$

Table 4. Table 4: Table of quadrupolar spherical tensor elements based on the final state reached by the transition. Each element of the table must be multiplied by α ​ ℏ ​ ω ​ k 2 ​ δ ​ ( E f − E i − ℏ ​ ω ) ​ ξ f 2 𝛼 Planck-constant-over-2-pi 𝜔 superscript 𝑘 2 𝛿 subscript 𝐸 𝑓 subscript 𝐸 𝑖 Planck-constant-over-2-pi 𝜔 superscript subscript 𝜉 𝑓 2 \alpha\hbar\omega k^{2}\delta(E_{f}-E_{i}-\hbar\omega)\xi_{f}^{2} where f 𝑓 f is the final state and ξ f subscript 𝜉 𝑓 \xi_{f} represents the integration of the radial parts of the wavefunctions.

final state	${\sigma }^Q(0,0)$	${\sigma }^Q(2,0)$	${\sigma }^Q(2,2)$	${\sigma }^Q(4,0)$	${\sigma }^Q(4,2)$	${\sigma }^{Qr}(4,4)$	${\sigma }^{Qr}(4,-4)$
$d_{z^2}$	$2/\sqrt{5}$	$-4/\sqrt{14}$	$0$	$12/\sqrt{70}$	$0$	$0$	$0$
$d_{x^2-y^2}$	$2/\sqrt{5}$	$4/\sqrt{14}$	$0$	$2/\sqrt{70}$	$0$	$1$	$1$
$d_{xy}$	$2/\sqrt{5}$	$4/\sqrt{14}$	$0$	$2/\sqrt{70}$	$0$	$-1$	$-1$
$d_{xz}$	$2/\sqrt{5}$	$-2/\sqrt{14}$	$-\sqrt{3/7}$	$-8/\sqrt{70}$	$2/\sqrt{7}$	$0$	$0$
$d_{yz}$	$2/\sqrt{5}$	$-2/\sqrt{14}$	$\sqrt{3/7}$	$-8/\sqrt{70}$	$-2\sqrt{7}$	$0$	$0$

Equations

$${\sigma }^D(\widehat{ϵ})={\sigma }^D(0,0)-\frac{1}{\sqrt{2}}(3{\cos }^2\theta -1){\sigma }^D(2,0)$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})={\sigma }^Q(0,0)\\ +\sqrt{\frac{5}{14}}(3{\sin }^2\theta {\sin }^2\psi -1){\sigma }^Q(2,0)\\ +\frac{1}{\sqrt{14}}[35{\sin }^2\theta {\cos }^2\theta {\cos }^2\psi \\ +5{\sin }^2\theta {\sin }^2\psi -4]{\sigma }^Q(4,0)\\ +\sqrt{5}{\sin }^2\theta [({\cos }^2\theta {\cos }^2\psi \\ -{\sin }^2\psi )\cos 4\varphi -2\cos \theta \sin \psi \cos \psi \sin 4\varphi ]{\sigma }^{Qr}(4,4)\end{array}$$

$$R=\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ 0& 0& 1\end{array}\right)$$

$$\widehat{ϵ}=\left(\begin{array}{c}\sin \theta \cos \varphi \\ \sin \theta \sin \varphi \\ \cos \theta \end{array}\right),\hat{k}=\left(\begin{array}{c}\cos \theta \cos \varphi \\ \cos \theta \sin \varphi \\ -\sin \theta \end{array}\right).$$

$$\frac{I_f}{I_0}=ϵ\left(\frac{\Delta \Omega }{4\pi }\right)e^{-a{\mu }_f{d}^{\prime }}\frac{\mu }{\mu -a{\mu }_f}\left[1-e^{-(\mu -a{\mu }_f)d^{\prime }}\right]$$

$$\frac{I_f}{I_0}=ϵ\left(\frac{\Delta \Omega }{4\pi }\right)(1-{\mu }_{f}a{d}^{\prime })\mu d^{\prime }$$

$$\sigma =\frac{\mu }{N}$$

$${\sigma }^D(\widehat{ϵ})={\sigma }^D(0,0)-\frac{1}{\sqrt{2}}(3{\cos }^2\theta -1){\sigma }^D(2,0)$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})={\sigma }^Q(0,0)+\sqrt{\frac{5}{14}}(3{\sin }^2\theta {\sin }^2\psi -1){\sigma }^Q(2,0)\\ +\frac{1}{\sqrt{14}}[35{\sin }^2\theta {\cos }^2\theta {\cos }^2\psi +5{\sin }^2\theta {\sin }^2\psi -4]{\sigma }^Q(4,0)\\ +\sqrt{5}{\sin }^2\theta [({\cos }^2\theta {\cos }^2\psi -{\sin }^2\psi )\cos 4\varphi -2\cos \theta \sin \psi \cos \psi \sin 4\varphi ]{\sigma }^{Qr}(4,4)\end{array}$$

$$⟨\sigma (l,m)⟩=\frac{1}{|G^{\prime }|}\sum _gϵ(g)\sum _{m^{\prime }=-l}^{m^{\prime }=l}\sigma (l,m^{\prime })D_{m^{\prime }m}^l(g)$$

$$\begin{array}{r}⟨{\sigma }^D(0,0)⟩=\frac{1}{8}\sum _gϵ(g){\sigma }^D(0,0)D_{00}^0(g)={\sigma }^D(0,0)\\ ⟨{\sigma }^D(2,0)⟩=\frac{1}{8}\sum _g\sum _{m^{\prime }=-2}^{m^{\prime }=2}ϵ(g){\sigma }^D(2,m^{\prime })D_{m^{\prime }0}^2(g)={\sigma }^D(2,0)\end{array}$$

$$\begin{array}{r}⟨{\sigma }^Q(0,0)⟩=\frac{1}{8}\sum _g{\sigma }^Q(0,0)D_{00}^0(g)={\sigma }^Q(0,0)\\ ⟨{\sigma }^Q(2,0)⟩=\frac{1}{8}\sum _g\sum _{m^{\prime }=-2}^{m^{\prime }=2}{\sigma }^Q(2,m^{\prime })D_{m^{\prime }0}^2(g)={\sigma }^Q(2,0)\\ ⟨{\sigma }^Q(4,0)⟩=\frac{1}{8}\sum _g\sum _{m^{\prime }=-4}^{m^{\prime }=4}{\sigma }^Q(4,m^{\prime })D_{m^{\prime }0}^4(g)={\sigma }^Q(4,0)\\ ⟨{\sigma }^Q(4,4)⟩=\frac{1}{8}\sum _g\sum _{m^{\prime }=-4}^{m^{\prime }=4}{\sigma }^Q(4,m^{\prime })D_{m^{\prime }4}^4(g)=\frac{1}{2}\left({\sigma }^Q(4,4)+{\sigma }^Q(4,-4)\right)=\Re {\sigma }^Q(4,4)\end{array}$$

$${⟨\sigma (l,m)⟩}_X=\frac{1}{n}\sum _{i=1}^n\sum _{m^{\prime }=-l}^{m^{\prime }=l}ϵ(g_i)⟨\sigma (l,m)⟩D_{m^{\prime }m}^l(g_i^{-1})$$

$$R_{eq}=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$$

$$R_{eq}^{-1}=\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right)$$

$$\begin{array}{r}{⟨\sigma (0,0)⟩}_X=\frac{1}{2}\sum _{i=1}^2⟨\sigma (l,m)⟩D_{m^{\prime }m}^l(g_i^{-1})=⟨\sigma (0,0)⟩\\ {⟨\sigma (2,0)⟩}_X=⟨\sigma (2,0)⟩\\ {⟨\sigma (4,0)⟩}_X=⟨\sigma (4,0)⟩\\ {⟨\sigma (4,4)⟩}_X=-⟨\sigma (4,4)⟩\end{array}$$

$$\cos (\varphi +\frac{\pi }{4})=\frac{\sqrt{2}}{2}(\cos \varphi -\sin \varphi ),\sin (\varphi +\frac{\pi }{4})=\frac{\sqrt{2}}{2}(\cos \varphi +\sin \varphi )$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})={\sigma }^Q(0,0)+\sqrt{\frac{5}{14}}(3{\sin }^2\theta {\sin }^2\psi -1){\sigma }^Q(2,0)\\ +\frac{1}{\sqrt{14}}[35{\sin }^2\theta {\cos }^2\theta {\cos }^2\psi +5{\sin }^2\theta {\sin }^2\psi -4]{\sigma }^Q(4,0)\\ -\sqrt{5}{\sin }^2\theta [({\cos }^2\theta {\cos }^2\psi -{\sin }^2\psi )\cos 4\varphi -2\cos \theta \sin \psi \cos \psi \sin 4\varphi ]{\sigma }^{Qr}(4,4)\end{array}$$

$${\sigma }^D(\widehat{ϵ})={\sigma }^D(0,0)-\frac{1}{\sqrt{2}}(3{\cos }^2\theta -1){\sigma }^D(2,0)$$

$${\sigma }^D(\widehat{ϵ})\propto \alpha \hslash \omega \left(\frac{2}{\sqrt{5}}-\frac{1}{\sqrt{7}}(3{\cos }^2\theta -1)\right){\xi }_{z^2}^2\delta \left(E_f-E_i-\hslash \omega \right)$$

$${\sigma }^D(\widehat{ϵ})\propto \alpha \hslash \omega \left(\frac{2}{\sqrt{5}}+\frac{1}{\sqrt{7}}(3{\cos }^2\theta -1)\right){\xi }_{x,y}^2\delta (E_f-E_i-\hslash \omega )$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})={\sigma }^Q(0,0)+\sqrt{\frac{5}{14}}(3{\sin }^2\theta {\sin }^2\psi -1){\sigma }^Q(2,0)\\ +\frac{1}{\sqrt{14}}[35{\sin }^2\theta {\cos }^2\theta {\cos }^2\psi +5{\sin }^2\theta {\sin }^2\psi -4]{\sigma }^Q(4,0)\\ -\sqrt{5}{\sin }^2\theta [({\cos }^2\theta {\cos }^2\psi -{\sin }^2\psi )\cos 4\varphi -2\cos \theta \sin \psi \cos \psi \sin 4\varphi ]{\sigma }^{Qr}(4,4)\end{array}$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})\propto \alpha \hslash \omega k^2\left(\frac{2}{\sqrt{5}}+\frac{1}{7\sqrt{5}}\left(35{\sin }^2\theta {\cos }^2\theta -4\right)\right){\xi }_{z^2}^2\delta (E_f-E_i-\hslash \omega )\end{array}$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})\propto \alpha \hslash \omega k^2\left(\frac{2}{\sqrt{5}}+\frac{1}{42\sqrt{5}}\left(35{\sin }^2\theta {\cos }^2\theta -4\right)+\sqrt{5}{\sin }^2\theta {\cos }^2\theta \right){\xi }_{xy}^2\delta (E_f-E_i-\hslash \omega )\end{array}$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})\propto \alpha \hslash \omega k^2\left(\frac{2}{\sqrt{5}}+\frac{1}{42\sqrt{5}}\left(35{\sin }^2\theta {\cos }^2\theta -4\right)-\sqrt{5}{\sin }^2\theta {\cos }^2\theta \right){\xi }_{x^2-y^2}^2\delta (E_f-E_i-\hslash \omega )\end{array}$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})\propto \alpha \hslash \omega k^2\left(\frac{2}{\sqrt{5}}-\frac{2}{21\sqrt{5}}\left(35{\sin }^2\theta {\cos }^2\theta -4\right)\right){\xi }_{yz}^2\delta (E_f-E_i-\hslash \omega )\end{array}$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})\propto \alpha \hslash \omega k^2\left(\frac{71\sqrt{5}}{168}-\frac{\sqrt{5}}{4}\cos 4\varphi \right){\xi }_{x^2-y^2}^2\delta (E_f-E_i-\hslash \omega )\end{array}$$

$$\begin{array}{r}{\sigma }^Q(\widehat{ϵ},\hat{k})\propto \alpha \hslash \omega k^2\left(\frac{71\sqrt{5}}{168}+\frac{\sqrt{5}}{4}\cos 4\varphi \right){\xi }_{xy}^2\delta (E_f-E_i-\hslash \omega )\end{array}$$

Full text

I Introduction

Titanium dioxide (\chTiO2) is one of the most studied large-gap semiconductor due to its present and potential applications in photovoltaics [1] and photocatalysis [2]. The increasingly strict requirements of modern devices call for sensitive material characterization techniques which can provide local insights at the atomic level [3, 4]. K-edge X-ray absorption spectroscopy (XAS) is an element specific technique, that is used to extract the local geometry around an atom absorbing the X-radiation, as well as about its electronic structure [5]. A typical K-edge absorption spectrum usually consists of three parts: (i) in the high energy region above the absorption edge (typically $>$50\text{,}\mathrm{eV}), the extended X-ray absorption fine structure (EXAFS), contains information about bond distances. Modelling of the EXAFS is rather straightforward, as the theory is well established [[5](#bib.bib5)]; (ii) The edge region and slightly above it ($<$50\text{\,}\mathrm{eV}) represents the X-ray absorption near edge structure (XANES), which contains information about bond distances and bond angles around the absorbing atom, as well as about its oxidation state. In contrast to EXAFS, XANES features require more complex theoretical developments due to the multiple scattering events and their interplay with bound-bound atomic transitions; (iii) The pre-edge region consists of bound-bound transitions of the absorbing atom. In the case of transition metals, the final states are partially made of $d$-orbitals. Pre-edge transitions thus deliver information about orbital occupancies and about the local geometry because the dipole-forbidden $s$-$d$ transitions are relaxed by lowering of the local symmetry. The \chTi K-edge absorption spectrum of anatase \chTiO2 (a-\chTiO2) exhibits four pre-edge features labelled A1, A2, A3 and B, while rutile \chTiO2 only shows three [6, 7]. Their assignment has been at the centre of a long debate, which is still going on, especially in the case of the a-\chTiO2 polymorph [8, 9, 10]. In this article, we use XAS linear dichroism at the Ti K-edge to assign the pre-edge transitions of a-\chTiO2 since this technique can provide the orbital content in the final state of the bound transitions with the support of ab initio finite difference method (FDM) calculations and spherical tensor analysis of the absorption cross-section.

Early theoretical developments to explain the origin of pre-edge features in a-\chTiO2 were based on molecular orbital (MO) theory [11, 12, 13] which showed that the first two empty states in a-\chTiO2 are made of antibonding $t_{2g}$ and $e_{g}$ orbitals derived from the $3d$ atomic orbitals of \chTi. Transitions to these levels have, respectively, been assigned to the A3 and B peaks while the absorption edge is made of \chTi $t_{1u}$ antibonding orbitals derived from \chTi $4p$ atomic orbitals. Although MO theory can predict the energy position of the transitions accurately, it cannot compute the corresponding cross-sections and does not account for the core-hole to which quadrupolar transitions to $d$-orbitals at the K-edge are extremely sensitive [9]. The corresponding transitions are usually red shifted by the core-hole and appear as weak peaks on the low energy side of the pre-edge. In a-\chTiO2, peak A1 contains a significant quadrupolar component [9], sensitive to the core hole, which explains the inaccuracy of MO theory to predict this transition. Full multiple scattering (FMS) is a suitable technique to treat large ensembles of atoms and obtain accurate cross-sections [12, 14, 10, 8]. From FMS calculations, a consensus has emerged assigning a partial quadrupolar character to A1, a mixture of dipolar and quadrupolar character with $t_{2g}$ orbitals to A3 and a purely dipolar transition involving $e_{g}$ orbitals to B [12, 15]. However, as correctly pointed out by Ruiz-Lopez [12], this simple picture of octahedral symmetry energy split t2g and $e_{g}$ levels becomes more complicated in a-\chTiO2 because of the local distorted octahedral environment (D2d symmetry) which allows local $p-d$ orbital hybridization [16]. In that case, the dipolar contribution to the total cross-section becomes dominant for every transition in the pre-edge region [17]. In addition, the cluster size used for the FMS calculations has a large influence on the A3 and B peak intensities showing that delocalized final states (off-site transitions) play a key role in the pre-edge absorption region [12]. Finally, the local environment around \chTi atoms in a-\chTiO2 is strongly anisotropic and \chTi-\chO bond distances separate in two groups of apical and equatorial oxygens which cannot be correctly described with spherical muffin-tin potentials as implemented in FMS. This limitation is overcome with the development of full potential FDM calculations such as finite difference method for near-edge structure (FDMNES) [18, 19, 20].

Empirical approaches have been used by Chen and co-workers [21] and Luca and co-workers [22, 23, 7] to establish correlations between the \chTi K pre-edge transitions in a-\chTiO2 and sample morphologies, showing that bond length and static disorder contribute to the change in the pre-edge peak amplitudes [21] and that the A2 peak is due to pentacoordinated \chTi atoms [22, 23, 7]. Farges and co-workers confirmed this assignment with the support of multiple scattering (MS) calculations [14]. The recent works by Zhang et al. [24] and Triana et al. [15] have shown the strong interplay between the intensity of pre-edge features and the coordination number and static disorder, in particular in the case of the A2 peak. However, the A2 peak is also present in the XAS of single crystals which suggests that the underlying transition is intrinsic to defect free a-\chTiO2. Clear evidence of the nature of this transition is lacking which is provided in this work.

The clear assignment of the pre-edge features of a-\chTiO2 is important in view of recent steady-state and ultrafast XAS [25, 26, 27] and optical experiments [28]. In the picosecond XAS experiments on photoexcited a-\chTiO2 nanoparticles above the band gap, a strong enhancement of the A2 peak was observed, along with a red shift of the edge [25]. This was interpreted as trapping of the electrons transferred to the conduction band at undercoordinated \chTi defect centres that are abundant in the shell region of the nanoparticles, turning them from an oxidation state of +4 to +3 [25]. The trapping time was determined by femtosecond XAS to be ca. $200\text{\,}\mathrm{fs}$, i.e. the electron is trapped immediately at or near the unit cell where it was created [26, 27]. Further to this, the trapping sites were identified as being due to oxygen vacancies (\chO_vac) in the first shell of the reduced \chTi atom. These \chO_vac’s are linked to two \chTi atoms in the equatorial plane and one \chTi atom in the apical position to which the biexponential kinetics (hundreds of ps and a few ns) at the \chTi K-edge transient was attributed [26, 29]. However, this hypothesis awaits further experimental and theoretical confirmation. In this sense, the assignment of peak A2 which provides the most intense transient signal in the pre-edge of a-\chTiO2 is a prerequisite.

In this article, we provide a detailed characterization of the steady-state XAS spectrum by carrying out a linear dichroism (LD) study of anatase \chTiO2 single crystals at the \chTi K-edge, accompanied by detailed theoretical modelling of the spectra. We fully identify the four pre-edge bands (A1-A3 and B) beyond the octahedral crystal field splitting approximation used in previous studies [30, 17]. Their dipolar and quadrupolar character is analyzed in detail as well as their on-site vs inter-site nature. The novelty resides in the quantitative reproduction of the experimental LD data with FDM calculations, the observation of the quadrupolar nature of peak A2 in agreement with theoretical predictions [31] and the corresponding assignment of peak A2 as originating from a quadrupolar transition in single crystals and from defect states in nanomaterials which accidentally trigger a blue shift of peak A1 in the region of peak A2. This delivers a high degree of insight into the environment of \chTi atoms, which is promising for future ultrafast X-ray studies of the photoinduced structural changes in this material.

II Experimental setup

II.1 Linear dichroism

The LD measurements are performed at the microXAS beamline of the SLS in Villigen, Switzerland using a double \chSi(311) crystal monochromator to optimize the energy resolution. Energy calibration is performed from the first derivative of the XAS spectrum of a thin \chTi foil. We used a moderately focused rectangular-shaped X-ray beam of $20\times 200\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ in horizontal and vertical dimension, respectively. The XAS spectrum is obtained in total fluorescence yield with a Ketek Axas detector system with Vitus H30 SFF and ultra-low capacitance Cube-Asic preamplifier (Ketek Gmbh).

The sample consists of a (001)-oriented crystalline a-\chTiO2 thin film of $30\text{\,}\mathrm{nm}$ thickness. Sample growth and characterization procedures are reported in the Supplementary Information (SI) §1. Figure 1 shows a schematics of the sample motion required for the experiment. The sample was placed in the center of rotation of a system of stages which allow for both sample in-plane rotation ($\phi$) and orthogonal out-of plane rotation ($\theta$). By convention, a set of Euler angles $(\theta,\phi,\psi)$ orients the electric field $\hat{\epsilon}$ and wavevector $\hat{k}$ with respect to the sample. $\theta$ measures the angle between $\hat{\epsilon}$ and the $[001]$ crystal direction ($\hat{z}$ axis of the sample frame) orthogonal to the surface. $\phi$ measures the angle between $\hat{\epsilon}$ and the sample rotation axis $\hat{x}$. In principle, a third angle $\psi$ is necessary to fix the position of the wavevector in the orthogonal plane to the electric field but here $\psi=$0\text{,}\mathrm{\SIUnitSymbolDegree}. The $\theta$ angles reported in the experimental datasets are with a maximum systematic offset of $\pm$0.2\text{\,}\mathrm{\SIUnitSymbolDegree} which comes from the precision setting up the $\theta=$0\text{,}\mathrm{\SIUnitSymbolDegree} reference from the sample half-clipping of the X-ray beam at grazing incidence. The precision of the rotation stage of $\pm$0.01\text{\,}\mathrm{\SIUnitSymbolDegree} is negligible with respect to this angular offset.

LD is usually studied with the sample rotated in the plane orthogonal to the incident X-ray beam ($\phi$-rotation) [32]. In this work, the novelty comes from the sample rotation around $\hat{x}$ ($\theta$-rotation) which provides the largest changes in the XAS. This rotation induces a change of X-ray footprint onto the sample surface. We clearly show that it does not introduce spectral distortions because the effective penetration depth of the X-rays through the material (between 97 and $580\text{\,}\mathrm{nm}$ across the absorption edge of a-\chTiO2 for the largest footprint at $\theta=$1\text{,}\mathrm{\SIUnitSymbolDegree}$$ used here [33]) is kept constant as the sample is much thiner than the attenuation length at the \chTi K-edge. Instead, the total amount of material probed by the X-rays changes due to the larger X-ray footprint when $\theta$ increases and a renormalization over the detected number of X-ray fluorescence photons is required. This is done with the support of the FDMNES calculations since a few energy points have $\theta$-independent cross-sections as previously reported on other systems [34, 35, 36, 37, 38, 39, 40] (vide infra)111For a spectrum measured well above the absorption edge, the atomic background absorption converges for any incident polarization and can also be used in principle to renormalize the spectra.. With this renormalization procedure performed at a single energy point ($4988.5\text{\,}\mathrm{eV}$), we could obtain a set of experimental points with $\theta$-independent cross-sections at the energies predicted by the theory confirming the reliability of the method. Hence, crystalline thin films with suitable thicknesses with respect to the X-ray penetration depth offer more possibilities to study LD effects than single crystals and prevent the usual self-absorption distortion of bulk materials when using total fluorescence yield detection [42].

III Theory

III.1 Recent developments in computational methods

Recently, there have been two main developments in the computation of XANES spectra. The first is based on band structure calculations (LDA, LDA+U,…), which compute potentials self-consistently with and without the core-hole before the calculation of the XAS absorption cross-section with a core-hole in the final state [17, 43]. This approach provides excellent accuracy but is limited to the few tens of eV above the absorption edge due to the computational cost of increasing the basis set to include the EXAFS. The second one, the FDMNES approach implemented by Joly [20, 19], overcomes the limitations of the muffin-tin approximation in order to get accurate descriptions of the pre-edge transitions especially for anisotropic materials. The recent theoretical work by Cabaret and coworkers combining GGA-PBE self-consistent calculations with FDMNES [17] concluded that in a-\chTiO2, peak A1 is due to a mixture of quadrupolar ($t_{2g}$) and dipolar transitions ($p-t_{2g}$), A3 to on-site dipolar ($p-e_{g}$), off-site dipolar ($p-t_{2g}$) and quadrupolar ($e_{g}$) transitions, while B is due to an off-site dipolar transition ($p_{z}-e_{g}$). These results, together with those of previous works are summarized in Table 1. However, experimental support to the pre-edge assignments is still lacking, and is provided in this work using LD XAS at the \chTi K-edge of a-\chTiO2 with the theoretical support of ab-initio full potential FDMNES calculations and spherical harmonics analysis of the XAS cross-section.

III.2 Finite difference ab-initio calculations

The ab-initio calculations of the XAS cross-section were performed with the full potential FDM as implemented in the FDMNES package [18, 19]. A cluster of $7.0\text{\,}\AA$ was used for the calculation with the fundamental electronic configuration of the oxygen atom and an excited state configuration for the titanium atom (\chTi: [Ar]3d14s24p1) as performed elsewhere [24]. We checked the convergence of the calculation for increasing cluster sizes and found minor evolution for larger cluster radii than $7.0\text{\,}\AA$ (123 atoms). The Hedin-Lundqvist exchange-correlation potential is used [44]. A minor adjustment of the screening properties of the $3d$ levels is needed to match the energy position of the pre-edge features with the experiment. We found the best agreement for a screening of 0.85 for the $3d$ electrons. After the convolution of the spectrum with an arctan function with maximum broadening of $1.5\text{\,}\mathrm{eV}$, a constant gaussian broadening of $0.095\text{\,}\mathrm{eV}$ is applied to account for the experimental resolution of the experiment and get the closest agreement with the broadening of the pre-edge peaks.

III.3 Spherical tensor analysis of the dipole and quadrupole cross-sections

Analytical expressions of the dipole and quadrupole XAS cross-sections ($\sigma^{D}(\hat{\epsilon})$ and $\sigma^{Q}(\hat{\epsilon},\hat{k})$, respectively) are obtained from their expansion into spherical harmonic components [32, 45]. The expressions of $\sigma^{D}(\hat{\epsilon})$ and $\sigma^{Q}(\hat{\epsilon},\hat{k})$ depend on the crystal point group which is D4h ($4/mmm$) for a-\chTiO2. The dipole cross-section is given by:

[TABLE]

and the quadrupole cross-section by:

[TABLE]

with $\theta$, $\phi$ and $\psi$ as defined in the \chTi site point group (D2d). $\sigma^{X}(l,m)$ with $X=D,Q$ is the spherical tensor with rank $l$ and projection $m$. $\sigma^{Xr}$ refers to the real part of the cross-section. The Euler angles $(\theta,\phi,\psi)$ in the experiment are referenced to the crystal frame which is rotated in the $(O,\hat{x},\hat{y})$ plane with respect to the Euler angles in the \chTi site frame. Consequently, the angles in equations 1 and 2 differ from the angles defined in Figure 1 by a rotation of $\phi$. In the \chTi site frame, the $\hat{x}$ and $\hat{y}$ axes are bisectors of the \chTi-O bonds while the crystal frame is along the bonds. The matrix $R$ to go from the site frame to the crystal frame is,

[TABLE]

In the following, the polarizations of $\hat{\epsilon}$ and $\hat{k}$ are given in the crystal frame. Consequently, the corresponding polarizations for the site frame are given by $\hat{\epsilon}_{S}=R^{-1}(\hat{\epsilon})$ and $\hat{k}_{S}=R^{-1}(\hat{k})$.

Although some terms of $\sigma^{D}(\hat{\epsilon})$ and $\sigma^{Q}(\hat{\epsilon},\hat{k})$ may be negative, the total dipolar and quadrupolar cross-sections must be positive putting constraints on the values of $\sigma^{D}(l,m)$ and $\sigma^{Q}(l,m)$. The electric field $\hat{\epsilon}$ and wavevector $\hat{k}$ coordinates in the $(\hat{x},\hat{y},\hat{z})$ basis of Figure 1 are given by:

[TABLE]

Hence the detail of the cross-section angular dependence in equations 1 and 2 requires the estimate of the spherical tensors $\sigma^{D}(l,m)$ and $\sigma^{Q}(l,m)$ as performed elsewhere [46]. The XAS cross-section measured experimentally is an average over equivalent \chTi atoms under the symmetry operations of the crystal space group. The analytical formula representing this averaged cross-section requires the site symmetrization and crystal symmetrization of the spherical tensors, which is provided in SI §7 and §8. From this analysis, we obtain nearly equal (up to a sign difference) crystal-symmetrized ($\braket{\sigma(l,m)}_{X}$), site-symmetrized ($\braket{\sigma(l,m)}$) and standard ($\sigma(l,m)$) spherical tensors. Assuming pure $3d$ and $4p$ final states in the one-electron approximation, analytical expressions are provided for $\sigma^{D}(\hat{\epsilon})$ and $\sigma^{Q}(\hat{\epsilon},\hat{k})$ whose angular dependence with $\theta$ and $\phi$ are given in Table 2. The full expressions of the cross-sections are provided in SI §8. In this paper, we analyze the angular dependence of the pre-edge peak intensities with $\theta$ and $\phi$ and assign them to specific final states corresponding to \chTi-$3d$ and/or $4p$ orbitals with the support of both FDM and spherical tensor analysis.

IV Results

The experimental evolution of the \chTi K-edge spectra with $\theta$ is depicted in Figure 2a. The spectra are normalized at $4988.5\text{\,}\mathrm{eV}$ where the cross-section is expected to be $\theta$-independent according to FDMNES calculations (shown by the leftmost black arrow in Figure 2b). From this normalization procedure, a series of energy points with cross-section independent of the $\theta$ angle appear in the experimental dataset, as predicted by the theory (black arrows in Figure 2a and 2b) showing the reliability of the normalization procedure. In the pre-edge, the amplitude of peak A1 is dramatically affected by the sample orientation. In the post-edge regions, significant changes are observed as well.

Ab-initio FDM calculations of the total XAS cross-section (including dipolar and quadrupolar terms) are presented in Figure 2b for the same angles of incidence $\theta$ as in the experiment. In the pre-edge region, the trends for peak A1 and A3 are nicely reproduced. The absence of peak A2 at first sight, partially originating from defects [7, 21, 22, 23], is due to our perfect crystal modelling in the FDM calculations. In the post-edge region, a good agreement is found especially for the isosbestic points. This shows that a strong LD remains well above the edge in this material.

The evolution of the spectra is also shown for a fixed incidence angle $\theta=$45\text{,}\mathrm{\SIUnitSymbolDegree} while the sample is rotated around $\phi$ (Figure [3](#S4.F3)a)222The normalization energy is at $4988.5\text{\,}\mathrm{eV}$. The changes in amplitude are significantly less than with $\theta$-rotation. We observe a minimal evolution for the amplitudes of the peak B and at the rising edge from $4971\text{\,}\mathrm{eV}$ while a larger effect is distinguished in the spectral region of peaks A1, A2 and A3. *Ab-initio* calculations with the same $\hat{\epsilon}$ and $\hat{k}$ orientations as in the experiment are depicted in Figure [3](#S4.F3)b. Only a weak evolution of the amplitude of the pre-edge features is expected and is essentially located in the region of peaks A2 and A3. The amplitude should reach its maximum for $\phi=$180\text{\,}\mathrm{\SIUnitSymbolDegree}$[$90\text{\,}\mathrm{\SIUnitSymbolDegree}$]$ which is inconsistent with the experiment. Instead the fitted evolution of the pre-edge peak amplitudes shows that A2 undergoes a 30% peak amplitude change whose angular variation is compatible with a quadrupolar transition (SI Figure 6a) while A1, A3 and B have a maximum amplitude evolution of 10% (within the fitting confidence interval) with no specific periodicity (SI Figure 7). The strong variation in A2 peak amplitude can be observed by the appearance of a pronounced shoulder for $\phi=$150\text{\,}\mathrm{\SIUnitSymbolDegree} which becomes smoother for $\phi=$180\text{,}\mathrm{\SIUnitSymbolDegree}$$. Consequently, the main evolution is due to peak A2 which explains the disagreement with the perfect crystal FDM calculations. It also shows the essentially dipolar content of peaks A1, A3 and B which provide circles in polar plots along $\phi$ (SI Figure 7) which is in agreement with the results obtained from $\theta$-scans (vide infra). A fit of the A2 peak with a $90\text{\,}\mathrm{\SIUnitSymbolDegree}$-periodic function shows that it may be assigned to the contribution of $d_{x^{2}-y^{2}}$ orbitals from the expected angular evolution by spherical harmonic analysis (SI Figure 6a,b). However, the $d_{x^{2}-y^{2}}$ density of states (DOS) in the region of peak A2 is negligible with respect to $d_{xy}$ and $d_{z^{2}}$ (vide supra) hence we rely on the more pronounced angular evolution with $\theta$ in the following to show the involvement of $d_{xy}$ orbitals in the formation of peak A2.

In order to describe the origin of the LD with $\theta$ and assign the pre-edge resonances, the projected DOS of the final states for the pre-edge and post-edge region is depicted in Figure 4 (we drop the term ”projected” in the following for simplicity). Due to the large differences between the DOS of $s$, $p$ and $d$ states, a logarithmic scale is used vertically and normalized to the orbital having the largest DOS contributing to the final state among $s$, $p$ and $d$ orbitals. For peaks A1, A3 and B, most of the DOS comes from $d$-orbitals while $s$- and $p$-DOS are comparable. However, due to the angular momentum selection rule, the spectrum resembles the $p$-DOS as witnessed by the similarity between the integrated $p$-DOS and the calculated spectrum (black line in Figure 4e). Importantly, peak A1 has only ($p_{x},p_{y}$) contributions meaning that this transition is expected to have a much weaker intensity when the electric field gets parallel to the $\hat{z}$ axis, in agreement with the $\theta$-dependence of its amplitude (Figure 2). The $d$-DOS at peak A1 involves dxz, dyz and d${}_{x^{2}-y^{2}}$ orbitals, among which the first two can hybridize with the ($p_{x}$,$p_{y}$) orbitals and relax the dipole selection rules. The dipolar nature of A1 is also seen from the monotonic increase of its amplitude from $\theta=$0\text{,}\mathrm{\SIUnitSymbolDegree} to $\theta=$90\text{\,}\mathrm{\SIUnitSymbolDegree}, inconsistent with a quadrupolar allowed transition with $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ periodicity. Following the same analysis, peaks A3 and B do not undergo a strong change in amplitude under $\theta$-rotation because $(p_{x},p_{y})$ and $p_{z}$ contribute similarly to the DOS for these transitions although FDM calculations show that A3 should evolve in intensity with $\theta$ due to a $\sim$20% larger DOS for $p_{z}$ than for $p_{x},p_{y}$ as experimentally observed. From the integrated $d$-DOS along $(x,y)$ and $z$ (SI Figure 9a), we notice the inconsistency between the peak amplitudes in the theory and the experiment, which shows that they are essentially determined by the $p$-DOS (SI Figure 9b).

For a more quantitative description of the dipolar and quadrupolar components in the pre-edge, we extracted the quadrupolar cross-section from FDM calculations. It is depicted as thin lines in the inset of Figure 2b. The quadrupolar contributions are limited to peaks A1 and A3 with a contribution in the spectral region of peak A2. At peak A1, the quadrupolar amplitude is maximum for $\theta=$0\text{,}\mathrm{\SIUnitSymbolDegree} and $\theta=$90\text{\,}\mathrm{\SIUnitSymbolDegree} and the total cross-section becomes mainly quadrupolar for $\theta=$0\text{,}\mathrm{\SIUnitSymbolDegree} while the quadrupolar component contributes $\sim$ 15% of the peak amplitude for $\theta=$90\text{\,}\mathrm{\SIUnitSymbolDegree}. From the development of the cross-section into spherical harmonics (Table 2), the dipolar transitions to $p_{x,y}$ final states are expected to vary as $-\cos^{2}\theta$ while transitions to $p_{z}$ vary as $\cos^{2}\theta$ plus a constant (see SI Figure 12a). The fitted evolution of the dipolar cross-section of peak A1 in the experiment and in the FDM calculations is compatible with a transition to $p_{x,y}$ (green line in Figure 5a, fitting details in SI §2). The quadrupolar component (red line in Figure 5a) is compatible with a transition to $d_{xz},d_{yz}$ due to its $-\sin^{2}\theta\cos^{2}\theta$ predicted evolution in agreement with the $d$-DOS at peak A1 (Table 2 and Figure 12b in SI). The comparison between the experimental and theoretical amplitudes of peak A1 (Figure 5a) gives an excellent agreement further confirming that the A1 transition is mostly dipolar to $p_{x,y}$ final states. Following the same analysis, it is more difficult to determine the dominant $p$-DOS contributing to the transitions at peak A3 and B due to the weak evolution of their amplitude with $\theta$.

As pointed out earlier, the quadrupolar cross-section has a doublet structure in the region of peaks A2 and A3 (inset of Figure 2b). The most intense of the two peaks at $\theta=$45\text{,}\mathrm{\SIUnitSymbolDegree} is in the spectral region of peak A2 where the transition involving defects is expected in a-\chTiO2 nanoparticles for instance. A closer look at the fitted evolution of the A2 amplitude with $\theta$ shows a quadrupolar evolution with maximum value at $\theta=$45\text{\,}\mathrm{\SIUnitSymbolDegree} (Figure 5b). This is in agreement with the expected angular evolution of $d_{z^{2}}$ and $d_{xy}$ final states from spherical tensor analysis (SI Figure 12b) which contribute to the DOS in the spectral region of peak A2 (Figure 4c). It indicates that although the amplitude of A2 is underestimated in the FDM calculation, the consensus that A2 originates from undercoordinated and disordered samples may be more subtle because of the involvement of a transition in the perfect crystal and is discussed in the next section.

From this combined experimental and theoretical analysis, we emphasize that consecutive peaks in the pre-edge of a-\chTiO2 are not simply due to the energy splitting between t2g and eg as previously invoked [17]. This splitting is more complicated than the usual octahedral crystal field splitting because of the strong hybridization between $p$ and $d$ orbitals in a lowered symmetry environment which affects the relative ordering of the transitions. The consistent results between experiment, FDM calculations and spherical tensor analysis show the reliability of the assignment provided in this work. Table 1 compares our results with previous assignments of peaks A1 to B.

V Discussion

V.1 Local versus non-local character of the pre-edge transitions

Pre-edge transitions can originate either from on-site (localized) or off-site transitions involving neighbour \chTi atoms of the absorber. Off-site transitions are dipole allowed due to the strong $p-d$ orbital hybridization [16]. This effect has been shown on \chNiO, an antiferromagnetic (AF) charge-transfer insulator, for which the \chNi K-edge transition to $3d$ orbitals of the majority spin of the absorber is only possible between \chNi sites due to the AF ordering [43]. Hence, to disentangle between the local or non-local character of the pre-edge transitions in a-\chTiO2, we performed FDM calculations on clusters with increasing number of neighbour shells starting from an octahedral \chTiO_6 cluster with the same geometry and bond distances as in the bulk. The results are shown in SI Figure 8 with two orthogonal electric field orientations along $[001]$ ($\theta=$0\text{,}\mathrm{\SIUnitSymbolDegree}) and $[010]$ ($\theta=$90\text{\,}\mathrm{\SIUnitSymbolDegree}).

The calculation for \chTiO6 (green curve) shows only A1 meaning that it is mostly an on-site transition. The absence of peaks A3 and B suggests that they are mostly non-local transitions in agreement with Ref. [17]. Increasing the cluster size to $4\text{\,}\AA$ includes the second shell of \chTi ions, which generates most of the A3 amplitude. This shows that, similarly to \chNiO, an energy gap opens between the on-site and off-site transitions to $3d$ orbitals of \chTi and that A3 is mostly dipolar and strongly influenced by the intersite $3d-4p$ hybridization. Peak B is missing for this cluster size which shows that it is due to a longer range interaction and can be reconstructed with a $5\text{\,}\AA$ cluster including the next shell of neighbor \chTi atoms. A similar trend in the local or non-local character of the pre-edge transitions is observed at the metal K-edge of $3d$ transition metal oxides which has to do with the degree of $p-d$ orbital mixing [30].

V.2 Origin of peak A2

The experimental $\phi$ and $\theta$ angular evolution of the A2 amplitude (Figure 5b and 6a in the SI) matches a quadrupolar transition, qualitatively consistent with the dominant quadrupolar cross-section obtained from FDMNES calculations in the region of peak A2 (Figure 2b, inset and Figure 3b). Recent calculations accounting for the electron-hole interaction in the Bethe-Salpeter equation have reproduced peak A2, although with an underestimated amplitude as in our FDM calculations [31]. Peaks A1 and A2 are found to exhibit their maximum amplitude when the electric field is parallel to the $(\textbf{a},\textbf{b})$ and c axes, respectively. This is in agreement with our measurement for peak A1 (Figure 5a) as a result of the coupling between the $3d$ states of \chTi with the $p_{x,y}$ DOS. For peak A2, we observe a dominant quadrupolar evolution with a deviation from the ideal behavior showing the presence of $p_{z}$ states which increase their contribution to the transition when $\theta\rightarrow$0\text{,}\mathrm{\SIUnitSymbolDegree}$$. Although both peaks A1 and A2 peaks show p-d orbital mixing, this mixing is clearly stronger for the A1 peak where the dipole contribution becomes dominant over the quadrupolar in contrast to the A2 peak. It shows that the amount of p-d orbital mixing differs for these transitions which can be explained by the $\sim$100 times lower $p_{z}$ DOS in the region of peak A2 than $p_{x,y}$ DOS in the region of peak A1 (Figure 4b). The underestimated amplitude of peak A2 in our calculation is likely due to the lack of explicit treatment of the electron-hole interaction which would improve the agreement of energy and amplitudes for peaks A1 and A2 without resorting to changes in screening constants of the $3d$ electrons as in our study. A parallel can be made between the energy splitting of peaks A1 and A2 containing quadrupolar localized components and the splitting of the bound excitons of a-\chTiO2 observed in the optical range where the $(\textbf{a},\textbf{b})$ plane exciton has a larger binding energy than the c exciton [48, 49, 28].

While we show that the presence of the A2 peak can be explained by the electronic structure of a-\chTiO2, a number of previous studies have concluded that A2 is related to lattice defects [7, 21, 22, 23, 25, 26, 29]. The question arises as to the connection between the A2 peak and the lattice defects, if any. Oxygen vacancies are native defects in a-\chTiO2 [50]. The occurrence of an oxygen vacancy in the vicinity of a \chTi atom will further lower the D2d symmetry and introduce $p-d$ orbital mixing in the pentacoordinated \chTi atom increasing the transition amplitude while broadening the transition due to the inhomogeneous contribution of the vacancy distribution [24, 15]. In order to check the effect of an \chO_vac on the XAS spectrum of a \chTi atom in the vicinity, ab-initio FDM calculations are performed at the \chTi K-edge of \chTi atoms with a doubly ionized \chO_vac (\chV_O^2+) at the apical or equatorial position in a supercell of 768 atoms. The calculations are performed with a bulk a-\chTiO2 4x4x4 superlattice structure from which one oxygen atom is removed in the center and neighbor titanium atoms are moved along the broken \chTi–\chO bond to simulate lattice relaxation. We have taken the local structural relaxation reported in another work with hybrid functional calculations where the titanium atoms move away from \chV_O^2+ by $0.509\text{\,}\AA$ in the equatorial plane and $0.109\text{\,}\AA$ in the apical position [51]. The results, depicted in Figure 6, show a chemical shift of A1 in the region between peaks A1 and A3 of the perfect lattice where peak A2 is expected, while peaks A3 and B remain essentially unaffected by the \chO_vac. The blue-shift of peak A1 is explained from the essentially local character of its final state involving $3d$ orbitals in the final state of the quadrupolar transition sensitive to the core-hole. The \chO_vac generates a redistribution of the electrons among the three nearest \chTi atoms which better screen the core-hole charge and lead to a blue shift of the transition. Since peaks A3 and B involve the first and second coordination shell of \chTi atoms, the effect of the \chO_vac is likely negligible on the corresponding final states. Peak A2 can be viewed as a peak A1 replica undergoing an \sim$$1\text{\,}\mathrm{eV} blue shift under the influence of an \chO_vac. Similar results have been obtained in rutile \chTiO2 for which doubly ionized oxygen vacancies introduced blue shift of the \chTi $3d$-DOS by $1\text{\,}\mathrm{eV}$ [52]. A similar effect is present in a-\chTiO2 at the O K-edge where the asymmetry of the so-called $t_{2g}$ and $e_{g}$ peaks with a tail on the high-energy side cannot be reproduced in the calculations with a bulk structure [53]. The amplitude of these peaks increases upon heavy ion irradiation compatible with the formation of more oxygen vacancies [54]. Hence, we find that the occurrence of a transition corresponding to undercoordinated \chTi atoms in the region of peak A2 is a coincidence. The experimental spectrum of a-\chTiO2 nanoparticles with defects would be a linear combination of the \chO_vac spectra (red and blue curves in Figure 6) and the spectrum of hexacoordinated \chTi atoms in the bulk (black curve in Figure 6) which depends on the amount of vacancy in the system. However, this study shows that peak A2 is expected to be present even in crystalline a-\chTiO2 nanoparticles because the intrinsic quadrupolar transition is likely dominant over the defect contribution. The large spectral weight transfer from peak A1 to a transition in the region of peak A2 for pentacoordinated \chTi atoms is fully compatible with our recent studies on photoexcited a-\chTiO2 nanoparticles [25, 26, 29]. We therefore conclude that the intensity enhancement at the peak A2 originates from an essentially quadrupolar transition in the regular lattice and from a spectral shift of peak A1 in the region of peak A2 for pentacoordinated \chTi atoms with an \chO_vac.

VI Conclusion

In summary, a complementary approach using experimental LD measurements at the \chTi K-edge of a-\chTiO2, ab-initio FDM calculations and spherical tensor analysis provides an unambiguous assignment of the pre-edge features. We show that A1 is mainly due to a dipolar transition to on-site hybridized $4p_{x,y}-3d_{xz},3d_{yz}$ final states which give a strong dipolar LD to the transition with a weak quadrupolar component from $(3d_{xz},3d_{yz},3d_{x^{2}-y^{2}}$) states. The A3 peak is due to a mixture of dipolar transitions to hybridized $4p_{x,y,z}-(3d_{xy},3d_{z^{2}})$ final states as a result of strong hybridization with the $3d$ orbitals of the nearest \chTi neighbour with a small quadrupolar component. The B peak is purely dipolar ($4p$ orbitals in the final state) and is an off-site transition (the electron final state is delocalized around the absorbing atom). The distinction between on-site and off-site transitions is possible using different cluster sizes in the FDM calculations. The LD is visible well above the absorption edge due to the strong $p$-orbital polarization in a-\chTiO2 which affects the amplitude of the EXAFS. Surprisingly, a quadrupolar angular evolution of peak A2 is observed for the first time with a narrow bandwidth showing that it is an intrinsic transition of the single crystal. A connection between the unexpectedly large experimental amplitude of this peak in nanoparticles is made with oxygen vacancies forming pentacoordinated \chTi atoms. Crude FDMNES calculations show that A2 may be viewed as a A1 peak undergoing a blue shift because of the change in the core hole screening due to \chO_vac’s. This explains the relatively intense A2 peak in amorphous \chTiO2 [24] or upon electron trapping at defects after photoexcitation of anatase or rutile \chTiO2 [29, 25, 26]. The unprecedented quantitative agreement provided in this work is made possible by the continued improvement of computational codes including full potentials [18, 19, 20] and the more accurate description of the core-hole interaction in Bethe-Salpeter calculations [55, 31]. Experiments are on-going to extend this work to rutile \chTiO2.

The present results and analysis should be cast in the context of ongoing ultrafast X-ray spectroscopy studies at Free Electron Lasers [56, 57]. For materials such as a-\chTiO2, the increased degree of detail that can be gathered from such sources was nicely illustrated in a recent paper by Obara et al. [27] on a-\chTiO2, showing that the temporal response of the pure electronic feature (at the \chTi K-edge) was much faster (\sim$$100\text{\,}\mathrm{fs}) than the response of structural features (\sim$$330\text{\,}\mathrm{fs}) such as the pre-edge and the above-edge XANES. The present work shows that by exploiting the angular dependence of some of the features, even up to the EXAFS region, one could get finer details about the structural dynamics, in particular, of non equivalent displacements of nearest neighbours.

Acknowledgements.

We thank Yves Joly and Christian Brouder for fruitful discussions and Hengzhong Zhang for providing the FDMNES input files. We also thank Beat Meyer and Mario Birri of the microXAS beamline for their technical support as well as the Bernina station staff of the SwissFEL for lending us the goniometer stage. This work was supported by the Swiss NSF via the NCCR:MUST and grants 200020_169914 and 200021_175649 and the European Research Council Advanced Grants H2020 ERCEA 695197 DYNAMOX. G. F. M. and C. B. were supported via the InterMUST Women Fellowship.

Appendix A Sample growth and characterizations

(001)-oriented epitaxial thin films of anatase \chTiO2 are fabricated by pulse laser deposition (PLD). The vacuum chamber has a base pressure of about $10^{-5}\text{\,}\mathrm{Pa}$. A KrF excimer laser (Lambda Physik LPX 300, $25\text{\,}\mathrm{ns}$ pulses, $\lambda=$248\text{,}\mathrm{nm}$$) was used to ablate a target material of \chTiO2 fabricated in our laboratory. Commercially available (100)-oriented \chLaAlO3 (LAO) single crystal ($10\times 10\times 0.5\text{\,}\mathrm{mm}$) were used as substrates. The target to substrate distance was set at $40\text{\,}\mathrm{mm}$ and the laser energy at the target was about $32\text{\,}\mathrm{mJ}$ on a spot area of about $1.6\text{\,}{\mathrm{mm}}^{2}$ which gives an energy density of about $2.0\text{\,}\mathrm{J}\text{\,}{\mathrm{cm}}^{-2}$. The films were grown under oxygen partial pressure of $5\text{\,}\mathrm{Pa}$. Platinum paste was used to provide the thermal contact between the substrate and the heating stage. The substrate temperature was about $610(\pm 10)\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$, as measured with a pyrometer pointing at a \chPt black spot near the substrate. X-ray diffraction (XRD) and X-ray reflectometry (XRR) (PANalytical X’pert Pro MPD with \chCu K$\alpha_{1}$ radiation at $1.540\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) analyses were used to investigate the crystalline structure of the films and for the calibration of the deposition rate, respectively. XRD shows the (001)-orientation of the deposited anatase \chTiO2 thin film (Figure 7a). Besides the (h00) reflexes of the substrate, only two diffraction peaks are visible identified to the (004) and (008) reflexes of anatase \chTiO2 (001)-oriented epitaxially on (100)-LAO. Interference fringes near the (004) diffraction peak are fitted to provide a film thickness of $31\pm 1\text{\,}\mathrm{nm}$ (Figure 7b). The Kiessig fringes observed in XRR provide a sample thickness of $32.8\pm 0.1\text{\,}\mathrm{nm}$ (Figure 7c) in agreement with XRD.

Appendix B Fitting of the pre-edge peak amplitudes

The pre-edge peak amplitudes were fitted in Matlab with a set of five gaussians for the experimental data and a set of four pseudo-Voight lineshapes for the calculated spectra. Overlap between the spectra and the fits are shown in Figure 8 for $\theta$-scans and Figure 9 for $\phi$-scans. Fittings of individual spectra (at a given angle) are shown in Figure 10 for $\theta$-scans and Figure 11 for $\phi$-scans. We noticed the presence of a weak and narrow peak on the high energy side of peak B around $4975.5\text{\,}\mathrm{eV}$ which does not belong originally to a-\chTiO2. Its amplitude is maximum for $\theta\sim$45-50\text{,}\mathrm{\SIUnitSymbolDegree}$$ and shows no variation under $\phi$-rotation (Figure 9a). It is likely due to a combination of Bragg-reflection from the \chLAO substrate and a-\chTiO2 which are present in this angular range. In the fittings, we have applied a mask in the region of this peak to focus on the main features intrinsic to a-\chTiO2.

The spectra amplitudes do not match between the experiment and the theory because the experiment deals with a number of photons detected by an APD in a given emission cone while the calculations provide absorption cross-sections. The detected fluorescence yield per incident photon depends on the linear absorption coefficient $\mu$ in a non-linear fashion [AlsNielsen:2011vn],

[TABLE]

where $\epsilon$ is the probability to emit a X-ray fluorescence photon, $\Delta\Omega$ the solid angle of detection, $\mu_{f}$ the linear absorption coefficient of the fluorescence through a path length $d^{\prime}$ inside the sample. In the thin film approximation, $(\mu-a\mu_{f})d^{\prime}\rightarrow 0$ such that the fluorescence yield becomes,

[TABLE]

The normalization at the isosbestic point accounts for the changes in $\Delta\Omega$ providing spectra proportional to $\mu$. The linear absorption coefficient is linearly related to the absorption cross-section $\sigma$ by,

[TABLE]

where $N$ is the atomic number density. Hence, a multiplicative coefficient needs to be applied to the experimental data after normalization to match the evolution of the theoretical absorption cross-section providing the experimental points depicted in Figure 5a of the main text.

Appendix C Evolution of the A2, A3 and B peak amplitudes with $\theta$ and $\phi$

Appendix D FDMNES calculations for different cluster sizes

Appendix E Integrated DOS along $x$ and $y$ axes

Appendix F XAS spectrum of a given site

The unit cell of a-\chTiO2 has two equivalent \chTi sites (actually four equivalent sites but connected with the $(1/2,1/2,1/2)$ translation). The calculation of the XAS spectrum with FDMNES at each site is shown in Figure 16 for $\theta=$45\text{,}\mathrm{\SIUnitSymbolDegree} and Figure [17](#A6.F17) for $\phi=$0\text{\,}\mathrm{\SIUnitSymbolDegree} and $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ which show that both sites provide the same spectra.

Appendix G Crystal-symmetrization of the spherical tensors

First, the spherical tensors are symmetrized in the \chTi site frame and then in the crystal frame taking into account the possible equivalent sites in the unit cell. The experimentally measured XAS spectrum is the average over the equivalent \chTi sites.

G.1 Derivation of the site-symmetrized spherical tensors of anatase in the site frame

The local point group of anatase is D2d (subgroup of the crystal point group D4h) which contains 4 pure rotations ($E$,$C_{2}(z)$,$C^{\prime}2(x)$,$C^{\prime}2(y)$) and 4 rotoinversions (the same as the pure rotations combined with the inversion). The angular evolution of the dipole and quadrupole cross-sections in this point group are the same as in the crystal point group D4h which gives for the dipole cross-section

[TABLE]

and for the quadrupole cross-section,

[TABLE]

The site frame has the $\hat{z}$ axis chosen along the elongated axis of the \chTiO6 octahedron and the $\hat{x}$ and $\hat{y}$ axes bisectors of the equatorial \chTi-O bonds to form a direct frame. The site-symmetrized spherical tensor $\braket{\sigma(l,m)}$ in the site frame is given by:

[TABLE]

with $|G^{\prime}|=8$ the number of rotoinversions and pure rotations in the site point group, $g$ a rotoinversion in the site point group, and $D^{l}_{m^{\prime}m}$ a Wigner-D matrix element. The parity factor is $\epsilon(g)=-1$ when the spherical tensor is odd and the symmetry operation $g$ contains the inversion or $\epsilon(g)=1$ otherwise. The Euler angles of the pure rotations are $E:(0,0,0)$, $C_{2}(z):(0,0,\pi)$, $C^{\prime}_{2}(x):(0,-\pi,\pi)$ and $C^{\prime}_{2}(x):(0,-\pi,0)$ while for the corresponding rotations combined with the inversion we get $(0,-\pi,\pi)$, $(0,-\pi,0)$, $(0,0,0)$ and $(0,0,\pi)$ which represent two times a set of four Euler angles. This provides the following symmetrized dipolar spherical tensors in the site frame:

[TABLE]

and the following quadrupolar spherical tensors,

[TABLE]

where we have used time reversal symmetry for $\braket{\sigma^{Q}(4,4)}$.

G.2 From site-symmetrized spherical tensors to crystal-symmetrized spherical tensors

The coset method is a powerful way to calculate the spherical tensors averaged over the crystal from the spherical tensor symmetrized over a single site which has been developed by Brouder and coworkers [46]. The crystal-symmetrized tensor $\braket{\sigma(l,m)}_{X}$ of a given site is obtained from the site-symmetrized tensor $\braket{\sigma(l,m)}$ by the operation

[TABLE]

with $n$ the number of cosets and $g_{i}$ a symmetry operation between the equivalent \chTi sites. Briefly, the subgroup of $I4_{1}/amd$ (group 141 with choice of origin number 1) with periodic translations has 8 symmetry operations which leave the \chTi site at $(0,0,0)$ invariant: $(x,y,z)$, $(-x,-y,-z)$, $(-x,y,z)$, $(x,-y,z)$, $(y,-x,z)$, $(-y,x,z)$, $(-y,-x,-z)$ and $(y,x,-z)$ which gives $n=|I4_{1}/amd|/|D_{2d}|=32/8=4$ equivalent sites. However, the translation $(1/2,1/2,1/2)$ gives the same orientation for the \chTi site at the center as the site at the corner and since they will give the same XAS spectra, we only need to consider 2 equivalent sites. The sites at $(0,0,0)$ and $(0,1/2,1/4)$ are considered which are connected to each other with the representative of the cosets $(x,y,z)$ and $(-y,x+\frac{1}{2},z+\frac{1}{4})$. The symmetry operation which allows going from one site to the other is defined by the matrix $R_{eq}$,

[TABLE]

with inverse rotation which needs to be considered in equation 13,

[TABLE]

which corresponds to the Euler angles $(0,0,-\pi/2)$ in ZYZ convention. We can now calculate the crystal symmetrized spherical tensors:

[TABLE]

accounting for the change of frame from the site to the crystal frame. These relations are very simple relations with only a sign change for the $\braket{\sigma(4,4)}_{X}$ matrix element from the non-symmetrized spherical tensor element. From these calculations, we also conclude that the two equivalent sites of \chTi considered in the crystal-symmetrization have actually the same spectra since they are rotated by $\pi$ around the $\hat{x}$ or $\hat{y}$ axis of the crystal which does not change the relative orientation of the $p$ or $d$ orbitals involved in the transitions. Consequently, crystal-symmetrized, site-symmetrized and standard spherical tensors can be used interchangeably (except $\sigma(4,4)$). We checked the equivalence between the \chTi sites XAS with FDMNES calculations and both dipole and quadrupole components measured under any $\theta$ (SI Figure 16) or $\phi$ angle (SI Figure 17) have the same cross-sections for both sites. The crystal-symmetrized spherical tensors are introduced in equations 8 and 9 to obtain the angular evolution of the dipole and quadrupole cross-section remembering that the angles present in these equations are referenced in the site frame. This leaves the angle $\theta$ unchanged because the $\hat{z}$ axis is the same in the site frame or in the crystal frame but the $\phi$ angle in the crystal frame is $\phi+\frac{\pi}{4}$ in the site frame. Consequently, the expression of the dipole cross-section is unchanged while the expression of the quadrupole cross-section is given by,

[TABLE]

and provide the expression for the quadrupole cross-section (only the last term undergoes a sign change),

[TABLE]

The expression of the spherical tensor elements is given in Table 4 assuming a given final state $3d$ orbital and provide the relative amplitude between the terms of $\sigma^{Q}(\hat{\epsilon},\hat{k})$.

Appendix H Spherical tensors of anatase

H.1 Final expression of the dipole cross-section

[TABLE]

H.1.1 Final state is pz

[TABLE]

H.1.2 Final state is px or py

[TABLE]

The normalized evolution of the dipole cross-section with $\theta$ is depicted in Figure 18a.

H.2 Final expression of the quadrupole cross-section as a function of $\theta$

[TABLE]

In the configuration of the experiment, $\psi=0$ and $\phi=\pi/2$. Hence the dominant term in the expansion is the third term proportional to $\sigma^{Q}(4,0)$ or $\sigma^{Qr}(4,4)$ depending on the final state orbital.

H.2.1 Final state is d${}_{z^{2}}$

[TABLE]

H.2.2 Final state is dxy

[TABLE]

H.2.3 Final state is d${}_{x^{2}-y^{2}}$

[TABLE]

H.2.4 Final state is dxz or dyz

[TABLE]

The normalized evolution of the quadrupole cross-section with $\theta$ is depicted in Figure 18b.

H.3 Final expression of the quadrupole cross-section as a function of $\phi$

In the configuration of this experiment $\theta=\frac{\pi}{4}$ and $\psi=0$.

H.3.1 Final state is $d_{z^{2}}$, $d_{xz}$, $d_{yz}$

Since these final states have $\sigma^{Qr}(4,4)=0$, they do not depend on $\phi$ according to equation 22.

H.3.2 Final state is d${}_{x^{2}-y^{2}}$

[TABLE]

H.3.3 Final state is dxy

[TABLE]

The normalized evolution of the quadrupole cross-section with $\phi$ is depicted in Figure 18b.